Report

Task 1

1.1 Derive the finite element matrices \(\mathbf{M}\), \(\mathbf{K}\), \(\mathbf{C}\) and \(\mathbf{R}\) from the strong form advection/diffusion/reaction equation. (2.5 points)

Task 2

2.1 In which part of the code are Dirichlet boundary conditions applied? Include the function, code line(s) and any parameters that need to be defined (0.5 point)

2.2 In which part of the code are Neumann boundary conditions applied? Include the function, code line(s) and any parameters that need to be defined (0.5 point)

2.3 What are the effective boundary conditions for the parts of the boundaries on which the simulation does not do anything? Motivate your answer with an observation about the simulation results. (1 point)

2.4 Give a mathematical expression for the boundary conditions that are applied on the different boundaries of the domain (0.5 point)

Task 3

3.1 What happens if you assume the contaminant is inert by setting \(r=0\)? Include any quantitative values that change (1 point)

3.2 What happens if the diffusivity is reduced with a factor 10 (use the original value for \(r\))? (1 point)

Task 4

4.1 If you want to code a simulate function to compute the hydraulic head distribution on the domain, which of the finite element functions in this notebook could you reuse? Also mention any modifications you would make, if any. (0.5 point)

4.2 How would you deal with the boundary conditions for this problem? (1.0 point)

4.3 In this notebook, we have used velocity vectors inside each element. After solving for \(h\) on the nodes, how would you compute the velocity at the center of a given element in the mesh? The correct answer here involves the finite element shape functions. (0.5 point)

By Frans van der Meer, Delft University of Technology. CC BY 4.0, more info on the Credits page of Workbook.